Collusion, Randomization and Leadership in Groups Rohan Dutta, - - PowerPoint PPT Presentation

Collusion, Randomization and Leadership in Groups

Rohan Dutta, David K. Levine and Salvatore Modica 1

The Question

• What happens when collusive “Mancurian” groups such as trade-

unions, political parties, lobbying organizations and so forth compete in a game?

• The basic setting is one of exogenous groups
• We might expect that: given the play of other groups each group

chooses the best strategy for itself

• This does not work as you might hope when the group faces

incentive constraints internally

• One of our proposed solutions – leaders with ex post evaluation –

also has applications to issues of coalition formation traditionally studied in the cooperative game theory literature 2

Overview

• players are exogenously partitioned into groups within which

players are symmetric

• given the play of the other groups there may be several symmetric

equilibria for a particular group

• if group can collude they will agree to choose the equilibrium most

favorable for its members

• this leads to non-existence
• we augment the model by introducing shadow mixing
• we show how these collusion constrained equilibria arise as the

limit of games with perturbed beliefs

• show equivalence to a leadership game with ex post evaluation
• builds on models used in mechanism design theory to study

collusion in auctions 3

A Motivating Example

three players first two players form a collusive group and the third acts independently theory: given the play of player 3, players 1 and 2 should agree on the incentive compatible pair of (mixed) actions that give them the most utility each player chooses one of two actions, C or D and payoffs given in bi- matrix form 4

Payoffs

player 3 plays C payoff matrix for the actions of players 1 and 2 is a symmetric Prisoner's Dilemma game in which player 3 prefers that 1 and 2 cooperate C If player 3 plays D the payoff matrix for the actions of players 1 and 2 is a symmetric coordination game in which player 3 prefers that 1 and 2 defect D 5

Equilibrium

probability with which player plays set of equilibria for players 1 and 2 given then D strictly dominant for both player 1 and 2 so they play D,D two equilibria, both symmetric at C,C and D,D three equilibria, all symmetric, C,C, D,D and a strictly mixed equilibrium 6

Optimal Collusion

no choice, they have to do D,D (remark: also the unique correlated equilibrium) get 6 at C,C equilibrium and strictly less than 6 at any other correlated within group equilibrium no ambiguity about the preferences of the group: they unanimously agree in each case as to which is the best equilibrium. group best response play D,D play C,C best response of 3 – never indifferent and always does the wrong thing group at D,D play D so at C,C no equilibrium 7

Does this make sense?

a small change in the probability of leads to an abrupt change in the behavior of the group but how can the group know so exactly? rather it makes sense that as the beliefs of a group change the probability with which they play different equilibria varies continuously versus the theory: player 1 and 2 with probability 1 agree that in the former case and in the latter case that perhaps it makes more sense to say that they agree that with 90% of the time in the former case and mistakenly agree that 10% of the time? 8

The Cheshire Cat

for the moment suppose that in that limit only the randomization will remain assume that randomization is possible at the critical point when and the incentive constraint exactly binds, the equilibrium “assigns” an arbitrary probability to C,C being the equilibrium if we have chance of C,C and D,D then 3 is indifferent and we have an equilibrium 9

The Exogenous Group Model

players and groups actions available for members of group are a finite set a fixed assignment of players to groups all players within a group are symmetric; utility of player is and invariant with respect to within group permutations of the labels of other players are mixed actions for a member of group , profiles of play chosen from this set represent the universe in which in-group equilibria reside each group is assumed to possess a private randomizing device

• bserved only by members of that group that can be used to coordinate

group play restrict to finite subset and consider only in-group equilibria for group in which all players choose the same action 10

Discussion

finiteness simplifies probability distributions over a continuous set it creates a complication because in-group equilibria may not exist in a finite set will use approximate equilibrium to take care of that now write 11

Collusion

groups collude but must respect incentive constraints group objective: maximize the common utility that they receive when all are treated equally 12

Incentive Slack and Shadow Mixing

strictly positive numbers measuring in utility units the violation of incentive constraints that are allowed gain function degree to which incentive constraint is violated gain is greater than then the group cannot choose gain is less than or equal to group may mix with any probability onto if it is at least as good as the best

• strict best response

shadow mixing/best response set Collusion Constrained Equilibrium 13

Incentive Compatible Games

If contains a relatively fine grid of mixtures there will be an -Nash equilibrium with a small value of strictly bigger than the group can find an action that is guaranteed to satisfy the incentive constraints to the required degree : regardless of the behavior of the

• ther groups there is always a

approximate equilibrium within the group. A game is incentive compatible if for all existence in incentive compatible games follows from basic continuity properties of the shadow best response correspondence 14

Random Belief Models and Equilibrium

given the true play

• f the other groups, there is a common belief

by group that is a random function of that true play An -random group belief model is a density function that is a continuous as a function of and satisfies these can be constructed by standard methods of convolutions; an explicit closed form involving the Dirichlet is given in the paper be any probability distribution over

• “best best” responses

measurable as a function of . . an -random belief equilibrium as an such that . 15

Random Belief vs Collusion Constrained

Theorem: Fix a family of -random group belief models, an and an incentive compatible game. Then for all there exist - random group equilibria. Further, if are -random belief equilibria and then is a collusion constrained equilibrium. 16

What Difference Do Collusion Constraints Make?

3C 3D independent players model: unique Nash equilibrium DDD (5,5,5) group ignores incentive constraints: unique outcome CCC (6,6,5) collusion constrained: group shadow mixes 50-50 CC and 3 mixes 50- 50 (4.75,4.75,2.5)

• notice that this is worse for everyone than the ordinary Nash

equilibrium of the game at 5,5,5

• hence “collusion constrained” - group cannot be stopped from

colluding and cannot credibly commit to not doing so

• “Olsonian interest groups”

mechanism designer with safe alternative of (4.9,4.9,4.9) 17

Leadership Equilibrium

group leaders serve as explicit coordinating devices for groups we do not want leaders to issue instructions that members would not wish to follow give them incentives to issue instructions that are incentive compatible by allowing group members “punish” their leader here has a concrete interpretation as the leader's valence: the higher the more members are ready to give up to follow the leader leaders give orders that must be followed, but are evaluated ex post 18

A non-cooperative game of leaders

Each group is represented by two virtual players: leader and evaluator with the same underlying preferences as the group members Each leader has a “big enough” punishment cost . The game goes as follows: Stage 1: each leader privately chooses an action plan conceptually these are orders given to the members who must obey the

• rders.

Stage 2: the evaluator observes the action plan of the leader of his own group Stage 3: the evaluator chooses a response Payoffs: if the evaluator chooses he receives utility ; if he chooses he receives utility . If the evaluator chooses the leader is deposed and gets . Otherwise the leader gets utility 19

Equivalence of Leadership Equilibria

Note that the leader and evaluator do not learn what the other groups did until the game is over. Theorem: In an incentive compatible game are sequential equilibrium choices by the leaders if and only if they form a collusion- constrained equilibrium 20

Alliances: An Example

the conformists prisoner's dilemma two symmetric groups with at least three players each players choose between two actions if all group action payoffs are individual preferences reflect a desire for conformity: an individual player gets the payoff determined by the common action minus a fixed strictly positive penalty if he fails to choose the group action any pure choice of action by the group is incentive compatible basically cooperative game theory 21

Exogenous Groups

each group has the dominant action of and the outcome is that this is what both groups do and all players receive but: why should not somebody who can speak to both groups point out the clear benefit to all from forming a single group and make them coordinate on under his leadership but: if this happens then why does not a member of, say, group propose that by separating from the common group and playing ? all members of group 1 would receive instead of if both groups do this, we are back to and joining the combined group seems attractive again 22

A Proposal

consider explicitly that there are leaders that recommend actions as before and make utility bids in an effort to form coalitions group members will choose the best bid require that bids be credible in the sense that the expected utility group members receive when they choose the best bid should in fact be at least the utility they were promised suppose there are three leaders:

• two group leaders with preferences inherited from their respective

groups, and a common leader who cares about the average utility

• f all members of both groups
• group leaders send offers only to their own group
• common leader sends offers to both groups

23

Analysis of the Example

in equilibrium the group leaders always recommend while the common leader always says - take this as given leaders may only bid utility of either or in case of tie follow the group leader no pure strategy equilibrium for reasons outlined above 24

Mixed Equilibrium

group leaders each bids with probability common leader bids with probability accepts an offer of from the common leader get common leader to be indifferent between the two bids given he will be evaluated ex post the expected utility received by the groups should be , since in that case both bids of and are equally accurate hence the extent to which he may be punished can be determined endogenously to make him indifferent between the two bids so similarly 25

Qualitative Properties

small the equilibrium approximately and equilibrium probability of cooperation is measures how attractive is defection relative to cooperation small the conflict between the groups is small, the common group forms with high probability and the groups cooperate most of the time large the conflict between the groups is large, the common group forms with low probability and the groups rarely cooperate. 26

A Model of Endogenous Coalitions

• as sequential equilibrium of a leadership game
• leaders characterized by valences that break ties and groups to

whom they can make offers

• evaluators evaluate accuracy of bids along with recommendations
• f actions
• a subset of the correlated equilibria of the underlying game

27

Equilibrium in the Conformists Prisoners Dilemma

assume that the grid starts at , has gaps of length and does not contain the points there is a strongly symmetric equilibrium in which we denote by the probability with which a group leader bids less than or equal to and by the probability with which a common leader bids less than

• r equal to

which in the limit as this has the same basic comparative statics with respect to as the simplified example 28

Group Leaders Who Can Talk

you have to be able to make credible offers – meaning you can only make offers to groups who can punish you suppose just group leaders, but they can be punished by either group so they can talk to both groups let be grid points closest to and respectively, consider the bids then each group accepts the bid of from the other group leader if a group leader tries to outbid the other leader by offering to his own group then the other group can get at most so he cannot offer the other group more than that means he loses the other group, and hence the other group will accept their own leader's bid and choose making a liar of the leader in the eyes of his own group 29

Conclusion

• the bidding model gives sensible looking equilibria with plausible

comparative static properties

• the leadership structure matters
• the mixed equilibrium has strong robustness properties
• the two group leaders equilibrium is overturned if there is a second

group leader – who can then outbid the first

• a key issue in applications is to understand which groups leaders

with different preferences can talk to

• if a group can choose a leadership structure then choosing a single

leader with the same preferences who can be punished by other groups makes sense

• choosing a single leader gives him commitment power, choosing

someone who can be punished by other groups enables him to negotiate on your behalf 30